1891
Fedorov and Schönflies independently classify crystallographic space groups showing that there are 230 of them.

1892Poincaré publishes the first of three volumes of Les Méthodes nouvelles de la mécanique céleste (New Methods in Celestial Mechanics). He aims to completely characterise all motions of mechanical systems, invoking an analogy with fluid flow. He also shows that series expansions previously used in studying the three-body problem, for example by Delaunay, were convergent, but not in general uniformly convergent. This puts in doubt the stability proofs of the solar system given by Lagrange and Laplace.

1893Pearson publishes the first in a series of 18 papers, written over the next 18 years, which introduce a number of fundamental concepts to the study of statistics. These papers contain contributions to regression analysis, the correlation coefficient and includes the chi-square test of statistical significance.

1896
The prime number theorem is proved independently by Hadamard and de la Vallée-Poussin. This theorem gives an estimate of the number of primes there are up to a given number, showing that the number of primes less than n tends to infinity as n/log n.

1899Lyapunov devises methods which provide ways of determining the stability of sets of ordinary differential equations.

1900Hilbert poses 23 problems at the Second International Congress of Mathematicians in Paris as a challenge for the 20th century. The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more. Many of the problems were solved during the 20th century, and each time one of the problems was solved it was a major event for mathematics.

1900Goursat begins publication of Cours d'analyse mathematique which introduces many new analysis concepts.

1900Levi-Civita and Ricci-Curbastro publish Méthodes de calcul differential absolu et leures applications in which they set up the theory of tensors in the form that will be used in the general theory of relativity 15 years later.